Introduction To Trigonometry. h 28m 37 o Calculate the height of the tree: ……………….

What Does Trigonometry Do ? Without trigonometry we wouldn’t have sailed the world, satellites in space wouldn’t know where they were and planes would be lost. Without trigonometry architects couldn’t design buildings and engineers couldn’t make cars, planes….. Without trigonometry millions of calculations made all over the world to keep things moving couldn’t be made.

Naming Sides Of Triangles. Consider the right angled triangle shown below: It has an angle of size x o marked in the right hand corner. xoxo The side opposite the angle you are looking at is called the opposite side. Opp’ for short. Opp’ The side opposite the right angle is called the hypotenuse. Hyp’ for short. Hyp’ The remaining side is called the adjacent. Adj’ for short. Adj’

The Ratio Of Two Sides. Look at the right angled triangle ABC below: AB C If AB = BC = 6cm what type of triangle is ABC. 6cm Isosceles. Calculate the ratio of opp’  adj’ for angle x o. xoxo Opp’ = Opp’ 6Adj’ = Adj’ 6 Opp’ Adj, = 6 6 =1 What size is must the angle x o ? x o = 45 o

Now consider the same calculation again for the triangle below: AB C xoxo 10cm We can see at once that the opposite divided by the adjacent is 1 and that the angle x o is still 45 o The same result will occur for all Right angled isosceles triangles.

The Tangent Of An Angle. AB C xoxo Opp’ Adj’ The ratio of opposite divided by adjacent is called the tangent of the angle x o. Normally written as tan x o for short. Key Result. For all right angled triangles the value of the tangent ratio will determine the size of the angle x o regardless of the size of the triangle.

Using The Tangent Ratio. Consider once more the triangle we started with: AB C 6cm Opp’ Adj’ xoxo To calculate x o on a calculator follow the steps below: On your calculator select the following buttons: INVTan -1 1X o = 45 o =

Calculating Angles Using The Tangent Ratio. We have now found how to calculate angles using the tangent ratio as the following example shows. Calculate the angle a o below. 11cm 16cm a o (1) Identify the opposite side. opp’ (2) Identify the adjacent side. adj’ (3) Write down the tangent ratio. (4) Substitute your values. (5) Divide the ratio giving your answer to 3d.p. (6) Use your calculator to find the angle a o to 1d.p. x o = 55.5 o

Further Example. Calculate the angle b o. bobo 16m 12m Opp’ adj’ x o = 36.9 o Always follow the routine !!

What Goes In The Box ? Find the size of each of the unknown angles below using the tangent ratio. ans: 37.9 o (3) 2.7cm 1.8cm c o 51.5 o 33.7 o 40 o (1) 14cm 18cm a o (2) 27cm 34cm b o (3) 29.1m 34.7m d o